1. Field of the Invention
The present invention relates generally to electrical filters and in particular to lowpass biquad filters.
2. Description of the Related Art
A lowpass filter is a filter for attenuating or damping the frequency band beyond a given frequency, also called the filter cutoff. Thus the amplitude response of a lowpass filter is different for frequencies either above or below the filter cutoff.
Lowpass filters, such as Butterworth, Chebychev or Bessel filters, have been used for years in the signal processing of signals in a reverberant or high clutter background. These high order filters present poles only and generally consist of several biquad filters, or biquads. A biquad filter is generally a filter with a two pole and two zero filter topology, i.e., with a second order transfer function in the s-domain both in the numerator or denominator. The poles and zeroes are directly linked to the elimination capacities of the biquad filter. Other types of biquad filters, as will be described later on, can include only poles, with a denominator of a biquadratic form.
Commonly used implementations of lowpass filters require a lot of operational amplifiers (OP-Amp) and other components, resulting in complex circuit architectures, high current consumptions and low noise performances. Noise specifications as low as −130 dBc/Hz, as required for example in GSM applications, are difficult to achieve. This is largely due to the contribution of the current sources and sinks generally implemented in the lowpass filters.
For low noise applications, a high order filter uses biquads with high Q factors. The Q factor of a biquad is related to the tangent of the biquad pole position (tan θ) by the equation: tan θ=√{square root over (4Q2−1)}. Q factor and tan θ are two ways of indicating the pole positions of a biquad filter, the higher their value is, the harder the circuit is to design, as explained here after.
A 2Nth order lowpass filter (N being a non-nil integer), for example a Butterworth filter, is made of 2N poles. All 2N poles have the same magnitude, but present different angles θ, and thus different values of tan θ. However the 2N poles are actually made up of N pole-pairs, known as complex conjugate pairs. The purpose of the biquad filter is to provide a single pole-pair (thus two poles). Thus, to build a 2Nth order Butterworth filter, N cascading biquad filters are needed, each providing two poles, or a complex conjugate pole-pair.
For instance, a 2nd order lowpass Butterworth filter with a cutoff frequency |P|, one biquad filter is used to provide a pole-pair of magnitude |P| and tan θ=1. A 4th order lowpass Butterworth filter with cutoff frequency |P| will require 2 biquad filters in series, one to provide a pole-pair of magnitude |P| and tan θ=0.414 and a second to provide a pole-pair of magnitude |P| and tan θ=2.414. A 6th order low-pass Butterworth filter with cutoff frequency |P| will use three biquad filters in series, one to provide a pole-pair of magnitude |P| and tan θ=0.268, a second to provide a pole-pair of magnitude |P| and tan θ=1, and a third to provide a pole-pair of magnitude |P| and tan θ=3.732.
As the order of the filter increases, more biquad filters are used, and the tan θ (or Q) values of the last biquad filter gets higher. This large value of tan θ can be problematic, as the ratio of component values is often related to tan2θ.